Tuesday, March 19, 2019

The GISS V3 land/ocean temperature anomaly rose 0.05°C in February. The anomaly average was 0.92°C, up from December 0.87°C. It compared with a 0.046°C rise in TempLS V3 mesh. I can't find Jim Hansen's monthly report on the web site, but his email linked to this page.

Like GISS, Moyhu is coming out with a V4 which will use GHCN V4. Actually, V3 could use it too, but it is time for a new version. GISS has marked theirs as beta - the monthly averages are here. V4 said the rise was only 0.01°C; Moyhu's V4 said the rise was higher, at 0.128°C. V4 of course has many more stations, and somewhat to my surprise, has xtensive reporting quite early; however, there is also extensive later data, and I'm still getting a feel for when it settles. My reporting is mixed; the main table shows V4, but the report and graphs are V3. This will change soon. I'll show below the anomaly maps for Moyhu and GISS V4 as well.

The overall pattern was similar to that in TempLS. W Canada and NW US were very cold, though SE US was quite warm.. Europe was warm, especially NE (Russia). Warmth also in Siberia and Alaska, and nearby Arctic. Australia was still quite warm. I'll note now that March so far is looking a whole lot warmer.

As usual here, I will compare the GISS and previous TempLS plots below the jump.

Sunday, March 10, 2019

TempLS is in transition from using GHCN V3 (which it has used for the last eight years) to the new GHCN V4. I'll report the V4 mesh results here, but for now, the Moyhu posted data is based on V3. I've been delayed a little because it is timely to also bring out a new version of TempLS (V4). Apart from anything else, the code needs reorganising to handle the bigger datasets smoothly. The Moyhu data page is still reporting values based on V3, as are the various pages with globe maps.

I worried about whether GHCN V4 would report as punctually as V3. But it seems to do better. Their version of 9 March showed 8779 stations reporting, compared with 10653 in January. But unlike V3, where countries report en bloc, here the reporting is spread, so no region is completely missing. I'll show a station map below.

TempLS mesh with V3 GHCN showed a larger rise, from 0.686°C to 0.779°C in February. This was more in line with the rise calculated with NCEP reanalysis. However, the V3 based infilled grid method was in agreement with V4 mesh, as was V4 infilled.

I'll write a post soon with a comparison of the whole V4-based history compared with V3. I'll also post a new version of TempLS which I have been using - I'll call that V4 too. It's mainly a restructuring to deal with the greater memory demands of GHCN V4, and also to include my new approach to handling spherical harmonics enhancement.

The TempLS mesh anomaly (1961-90 base) was 0.746deg;C in February vs 0.708°C in January. That makes it the third warmest February in the (V4-based) record.

The main feature of the month was rather severe cold over western Canada and adjacent US, but with warmth just a little to the North in the Arctic and Alaska. Europe was also very warm. There were patches of cool in the Sahara and western China.
Here is the temperature map.

Monday, March 4, 2019

The Moyhu NCEP/NCAR index rose from 0.296°C in January to 0.393°C in February, on a 1994-2013 anomaly base. That makes it the warmest month since May 2017. The warmth came mainly in a spurt at the end of the month, which is continuing.

It was quite cold in W Canada and the US, although warm in Alaska and the SE USA. Warm in Europe, cold in the Sahara, and mostly warm in Antarctica, with the Arctic mixed.

Saturday, February 16, 2019

Update With this post I initially misread the GISS data file, taking values from the end of the row which are period averages, rather than the correct monthly averages for November and December, as ErikGBL noted in comments. It actually made only a small difference, but I have fixed it. I first had a small rise from December, now a small drop.

The GISS land/ocean temperature anomaly fell 0.01°C in January, which followed a 0.11°C rise the month before. The January anomaly average was 0.88°C, down from December 0.89°C. It compared with a 0.034°C fall in TempLS mesh . Jim Hansen's report is here.

The overall pattern was similar to that in TempLS. The US was actually fairly warm overall, despite the cold snap at the end of the month. Canada's Arctic islands were very cold, but a big band of warmth in NW Canada/Alaska, and another from E Siberia (which also had a cold snap) through to the Caspian region. SE Australia was hot.

As usual here, I will compare the GISS and previous TempLS plots below the jump.

Friday, February 15, 2019

I have written a lot about two topics - anomalies in general, and spatial averaging of temperature anomalies. Both are headings in the Moyhu interactive index. My main point on anomalies as used in temperature statistics is that they play an essential role in reducing inhomogeneity, thus making the inevitable sample use more reliable.

But I think they have a wider use. The general pattern is that a field, temperature T, say, is partitioned:
T=E+A
where E is some kind of expected value, and A is an anomaly. That sounds vague, and there are many ways you could make such a partition. But it becomes clear when associated with the use made of it. The purpose is often to calculate some linear function of T, say L(T). Here L will be some kind of average. But suppose L is an ideal, and we actually have to rely on an approximate version L1 (eg coarse sampling).

So L(T) is approximated by L1(T)=L1(E)+L1(A). Now the objective is to choose E in such a way that

L1(E) can be replaced by L(E), which is more accurate for that part

L1(A) loses some of the error that marred L1(T)

I will explain with three examples here, of which the third will be developed in detail, because it bears on my long-term project of more accurately calculating surface temperatures:

Temperature anomalies for global average (GISS). Here the station temperatures have some kind of mean of past values subtracted. It is usually the mean of some fixed period, eg 1961-90. Here is a typical post in which I explain why absolute temperatures are too inhomogeneous to average with the sample density we have, which the anomalies thus constructed are homogeneous enough, if only because they have approximately the same expected value (0). But in terms of this structure, they don't quite fit. The component E (climatology) that is removed is not calculated with a more accurate estimator, but is discarded, with only the anomaly average reported. But the mode of partition is still important, to be sure that the discarded part does not carry with it variation that should have been attributed to A. Using a consistent base period is one way of ensuring that.

Another example is one I developed in a post at WUWT. This was about the time samples, for a single station, that go into forming a monthly average. An objection sometimes raised that characterising that by two values per day (max and min) "violates Nyquist". Nyquist doesn't really tell us about this kind of sampling, but I looked at the error introduced in sampling regularly a few times a day (eg every 12 hours). This does create aliasing of some harmonics of the diurnal frequency to zero, which adds spurious amounts to the monthly average. I showed that this is a limited error, but which can be reduced with a T=E+A partition. Now E is an estimate of the regular diurnal cycle, based on fine (eg hourly) sampling of a few reference years. That takes most of the diurnal variation out of A, and so sparse sampling (L1) aliases very little to zero, and causes little error. The fine operator (L) can be applied to the E component.

The third example, to be developed here, is for the monthly spatial integration of anomalies. I'll describe it first rather generally. Suppose T is a variable to be integrated over any domain, and L1 is an approximate integration operator. Then suppose E is a Fourier fit with the first n orthogonal functions, where the inner product is integration with L1.

Then E contains the low-frequency terms which contribute most to the integral, leaving A with just high frequency fluctuations. These may be of lower amplitude that the whole, but more importantly, they will make little contribution to the low pass integration operator. This is still true for the approximate L1, since the basic reason for the low-pass is cancellation.

In fact, by the Fourier process, L1(A) is exactly 0, since A is orthogonal to the fitting functions, of which the lowest is a constant. So if the improved integration operator L is just exact integration, the varying fitted terms will make zero contribution to the integral, except for the constant term. The improved integral is just the integral of that constant. The improvement comes because the approximate integral of those low frequency components is replaced by the exact.

Integration methods for the sphere

I have developed four main methods for averaging temperature anomalies via numerical integration on the sphere, which I have incorporated into my monthly program TempLS, with results regularly reported here (click TempLS button). They are described in greatest detail here, with some further ideas. The methods described are

Simple gridding - just assign to each cell of a lat/lon grid the average of the anomalies in the cell, and then us ethe area-weighted average of the cells with data. This is basically what is done in HADCRUT

Infilled gridding - where cells without data are assigned a value derived from neighbouring cells in some way, and then the area-weighted average of all cells can be used. With the assignment done by kriging, this is the Cowtan and Way improvement of HADCRUT.

Finite element integration using an irregular triangular mesh. This is my preferred method, as it assigns to each point on Earth the interpolated value from the three nearest data points for each month.

Spherical Harmonics. I'll now talk more about that. In its original form, I did a least squares regression of the temperature with a set of spherical harmonics, and took the lowest coefficient as the integral.

The final method, using Spherical Harmonics (SHs), is not like the others, in that it doesn't introduce a subdivision of the area. But I now think it is best seen as not an independent method, but as an improvement applicable to any method. To just write out the regression maths, if L1(xy) is the scalar product, and H is the vector of fitting functions (SH), then the regression coefficients of the fit are

b = L1(H⊗H)-1L1(H⊗T), where ⊗ is the outer product from the weighted integration

and so b0L1(H0) is the average of T

I generally denote L1(H⊗H) as HwH, since L1 is just a weighted sum with weights w (this is the architecture of TempLS). The functions H are orthogonal wrt exact integration L, so L(H⊗H) would be a unit matrix. The approximate version deviates from this, and a figure of merit is the condition number of HwH. This is kept down if the integration method L1 is good.

In addition to the averaging methods listed above, I should add a very primitive one - the simple, unweighted average. This is generally regarded as too unreliable for use, but my original SH method, which actually performed very well, was basically the improvement of this method. You can of course expect better outcomes from improving better methods, and as I shall show, this happens. However, the relative improvement is much less. The reason is that the improvement comes from the improved integration of the SH components, and with, say, the mesh method, the finite element integration was already very good.

More on Spherical Harmonics

I described the functions here, with visualisation here, and a movie here. There are two parameters L and M; it is natural to think of M as determining the number of oscillations, with L varying from 0 to M (for fixed M) going from purely latitude oscillation to purely longitudinal. In the latter case (L=M), there are M periods of the sinusoidal variation around the equator. For SHs of order up to M, there are (M+1)2 functions, and these are the groupings that I will test below.

Tests

I looked at various means of anomalies for each of the months of 2018. I used the anomalies calculated by the mesh method; other integration schemes give slightly different values, but I don't think that changes the issues of integration. For each month and order M of SH, I calculate the mean for each of the four methods

No weighting (simple average)

Simple grid with no infill

Infilled grids

Integrating on irregular triangular mesh.

The value for M=0 is the unimproved value. Here is a plot of results. You can see the different months by using the buttons at the bottom:

You can see that the better methods, mesh and infilled grid, start out usually close together, with often the simple average furthest away. But then, as the order M incrases, they all converge for a while. Eventually this gets ragged as the higher order SH's are less well integrated by the integration scheme. The direct cause is ill-conditioning of the HwH matrix. I'll show here a similar plot of log10 of this condition number for each case.

There is an earlier discussion of SH improvements here. It shows (for Jan 2017) the sum of squares improvement, but more relevantly, a detailed map of the residuals. It shows how, after removing up to M=10, there are still spatial patterns in the residuals, some of which look like higher order harmonics.

Discussion

As you look through the months, the blue mesh line is usually fairly stable, at least up to about M=10. This indicates that the removal of SH components has little effect, because the mesh integration was already good, and replacing with exact integration makes little difference. The other lines start out separated; the black unweighted average often a long way away. But they converge toward the blue line, which says that the SH separation is giving better results.

The condition numbers also give a good quantification of the merit of the integration schemes. mesh is markedly better than infilled grid, which is in turn better than simple grid weighting, which in turn is better than no weighting at all. Better conditioning has the practical effect that mesh integration can be used to approximate temperature anomalies with a SH fit (as done monthly here) to higher order, and better resolution, than other methods.

An anomaly in approximation behaviour is February, in which there was convergence toward the blue, but that showed a gradual increase rather than stability after about M=5. I have looked into that; I don't really know what is different about the February mesh that would cause that. The meshes can be seen here. It is true that the February mesh does do a less accurate job with integrating the harmonics, as can be seen from the top row of HwH. I can't track that down to a visible difference in the mesh. The condition number for February rises earlier than it does for March, although not so different from January.

I had thought that, although subtracting fitted SHs did not improve mesh integration here, it might do so if stations were fewer. I tried that in conjunction with my culling process. However, no improvement was found there either. The reason seems to be that when stations are too sparse for SHs to be well integrated, so there is room for improvement, they are also too sparse to allow the fit to be identified. There just aren't enough degrees of freedom.

Conclusion

The idea of separating out an "expected" component prior to integration, which can then be integrated with a more accurate operator, has general application. In the case of spatial integration of monthly anomalies on the sphere, it gives a radical improvement of inferior methods. It does not do much for the better mesh integration, but the different behaviour quantifies the merits order of the various schemes.

Although the mesh method seems to be at least as good as SH upgrades of grid methods, the latter is faster if meshes would have to be calculated. Simple gridding is fast, as is the SH fitting. However, on my PC even full meshing takes only a few minutes. So I have upgraded the "SH method" in TempLS to be the enhancement of the mesh method, since I can generally re-use meshes anyway.

Thursday, February 7, 2019

The TempLS mesh anomaly (1961-90 base) was 0.704deg;C in January vs 0.737°C in December. That makes it the fourth warmest January in the record.

Land temperatures were down, but SST rose a little after two months of decline. Despite the late cold snap in the mid-west, the US was not particularly cold, although the Hudson's Bay region of Canada was. The Sahara was cool too, but there were big warm patches in Siberia, the Caspian region, Alaska, and, notably, Australia.
Here is the temperature map. As always, there is a more detailed active sphere map here.

Tuesday, February 5, 2019

Update In comments, Bryan Oz4caster pointed out that the results I posted were wrong, following the shutdown and recovery. So I checked, and found the problem - I was calculating correct numbers, but posting numbers for corresponding days in 2015.Oddly, the corresponding month figures for 2015 showed almost exactly the same changes. Nov to Jan 2015 0.106 0.212 0.209

Nov to Jan 2019 0.176 0.286 0.296

So I haven't had to change the headline, but I've fixed the figures in the text. The map and discussion based on it were correct.

The Moyhu NCEP/NCAR index was delayed by the US government shutdown, and then by some confusion after an apparent delay in finalising 2018 (which my system did not handle well - code being revised). So I'm reporting both December and January. There were ups and downs in late 2018, leaving November a cold month at 0.176°C. So the rise in December to 0.286°C was more of a return to normal, and that continued with the 0.296°C in January.

One remarkable feature in January was a band of warmth starting from the Kimberley region of W Australia and extending through the continent to New Zealand and beyond. This was indeed a very warm month in Australia's inland. It also shows the cold plume from Canada's Arctic islands through the US mid-west, though there is another band of warmth from Alaska to California. There was also a broad band of warmth from China through to Arabia and central Africa, although the Sahara and around the Mediterranean were cold. Polar regions tended to cool.

The BoM ENSO Outlook was downgraded to Watch, although some models predict a reappearance.

Wednesday, January 30, 2019

The TempLS mesh anomaly (1961-90 base) was 0.748deg;C in December vs 0.628°C in November. That continues the recent up-down alternation, and makes it the third warmest December in the record (near tie with 2017 for 2nd). The GHCN results were delayed by the US government hiatus.

Significantly, for the second month in a row, SST was down, after a few months of rise. But land temperatures were generally higher. The big contributors to this were N America and the Arctic. Almost everywhere was warm, including Europe and Australia. A cool patch in mid-Siberia/Mongolia.
Here (from here) is the plot of relative contributions to the rise (ie components weighted by area):
Here is the temperature map. As always, there is a more detailed active sphere map here.

Saturday, January 12, 2019

Usually I would have my analyses out by now, but as explained, the data I need is not available yet due to the US government shutdown. I'll post, with 2018 discussion, when they are available. However, the ECMWF, via Copernicus, is not subject to that shutdown, and their reanalysis data is out. A CSV listing of monthly global and European surface average is here. Relative to 1981-2010, November was 0.430°C; December was 0.468°C.

A description of relevant ECMWF data on average monthly temperature is here.

I won't try to summarise the features of the month; ECMWF has a very detailed description in their December report. I'll just show their map, which includes Europe too:

The ECMWF also reports on the year's result; 2018 was the fourth warmest year in the record, after 2016, 2017, 2015. Details and links here. Most surface records including TempLS will report the same ranking.

Tuesday, January 1, 2019

Wishing all readers the best for 2019, even if it doesn't look so promising right now.

Usually about this time I am gathering data for articles about 2018. But the US government shutdown has now closed down most NOAA sites, and it doesn't look like ending anytime soon. The first affected will be the NCEP/NCAR reanalysis index. But GHCN a,d ERSST are out too, so that cuts out TempLS, on which I would normally report by about the 8th. So fingers crossed.

The last NCEP/NCAR data was from 23rd Dec.AT that stage, there had been a big warm peak, and DEcember was looking to be a very warm month. However, that was passing, so the end result is likely to be cooler, althoug still a lot warmer than November.